Conventional frequentist statistical methods are based on very strong prior assumptions, disguised as fixed sampling distributions and model forms. These assumptions are routinely violated by real research settings, which are filled with bias sources, and by the users, who select models and variables based on the data; hence conventional methods have no justification in practice. Bayesian and related procedures allow use of more realistic assumptions, especially by incorporating bias sources into the analysis. The obstacles to their use have been computational and psychological. These procedures can now be approximated with standard software, however. Psychological resistance centers on repeating the falsehood that conventional methods are objective. Although conventional statistics are accepted, ritualized, and entrenched, they are not objective, in that they do not reflect directly the world outside of our minds. In observational settings they are no more than deductions from thought experiments of the form "If a perfect randomized trial generated these data, then the effects ought to be in this range". Their evaluation has been based almost entirely on how accurately they move from their premises to their deductions - i.e., pure mathematics. But statistical procedures need to be evaluated based on evidence that they enhance accuracy of judgments about the real world, not just that they satisfy mathematical criteria. The track record of conventional methods in observational epidemiology is poor: they encourage excessive certainty and false inferences. Subjective Bayesian methods allow one to perform a wider range of thought experiments, including a most crucial one: "Given that I think these data came from something far removed from a randomized trial or random sample, with problems in this sort of range, how should these data affect my judgments about the effects of interest?" Case studies suggest that Bayesian-like methods can do much better than conventional methods because they force one to reveal the prior prejudices in these inferences. This revelation makes them more easily seen for what all inferential methods are: tools to aid rational subjective judgment.